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International Economics, Robert A. Mundell, New York: Macmillan, 1968, pp. 298-317.

Hicksian Stability, Currency Markets, and the Pure Theory of Economic Policy 1,2

Robert A. Mundell

The stability analysis introduced by Hicks has been one of the most successful failures in economic theory. Originally developed to integrate statical and dynamical general equilibrium theory, it was used by Hicks ([24], p. 62) as a bridge between dynamics and comparative statics:

The laws of change of the price system, like the laws of change of individual demand, have to be derived from stability conditions. We first examine what conditions are necessary in order that a given equilibrium system should be stable; then we make an assumption of regularity; that positions in the neighbourhood of the equilibrium position will be stable also; and thence we deduce rules about the way in which the price-system will react to changes in tastes and resources.

But the stability conditions were not founded on an explicitly formulated dynamic system. Hicks had defined stability along conventional lines:

In order that equilibrium should be stable, it is necessary that a slight movement away from the equilibrium position should set up forces tending to restore equilibrium....

but concluded that stability requires that

. . . a rise in price makes supply greater than demand, a fall in price demand greater than supply.

This is indeed the stability condition corresponding to a dynamic system in which excess demand causes a rise in price, but, as Samuelson pointed out, the proposition is not explicitly derived as the condition of convergence of such a dynamical system. The problem may be considered trivial in the case of a single market,3 but it raises difficulties in analysis and interpretation in the case of multiple exchange (exchange of more than two commodities), as Hicks foresaw ([24], p. 66):

What do we mean by stability in multiple exchange ? Clearly, as before, that a fall in the price of X in terms of the standard commodity will make the demand for X greater than the supply. But are we to suppose that it must have this effect (a) when the prices of other commodities are given, or (b) when other prices are adjusted so as to preserve equilibrium in the other markets?

To resolve the difficulty Hicks introduced his concepts of perfect and imperfect stability. He noted first that it was necessary to distinguish a series of conditions-that a rise in the price of X will make supply greater than demand (1) all other prices being given, (2) allowing for the price of Y being adjusted to maintain equilibrium in the Y market, (3) allowing for the prices of Y and Z being adjusted, and so on, until all prices have been adjusted. He then defined as imperfectly stable a system in which a rise in price of a commodity causes excess supply for the commodity after all repercussions are allowed for, and as perfectly stable a system in which a rise in price causes excess supply regardless of how many other prices are adjusted to attain equilibrium values in their respective markets.4

Samuelson [88] criticized Hicks' concept of stability on the grounds, as stated above, that "stability conditions are not deduced from a dynamic model, except implicitly." Hicksian stability is not equivalent to "true" dynamic stability, and the Hicks conditions are neither necessary nor sufficient for the convergence of the dynamical system implicit in Hicks' dynamics, that is, the dynamic system Samuelson postulated as the "natural extension" of the Walrasian system. True dynamic stability requires that the roots of the characteristic equation of the dynamic system have negative real parts, and this requirement is not equivalent to the Hicks conditions.5

Samuelson's criticism undermined the logic of Hicks' method. But the conditions of stability produced by that method retained an important place in the literature. Samuelson had already observed that the Hicks conditions were equivalent to the conditions of "true" dynamic stability in the symmetrical case. Metzler showed that they were equivalent in the case of gross substitutes, and also necessary (but not sufficient) for stability to be independent of the speeds of adjustment (Metzler [62]).6 Morishima proved that they were equivalent for certain classes of complements. They are also sufficient conditions for convergence of any nonoscillating system, since Hicksian (perfect) stability implies the absence of positive real roots. They are, moreover, conditions that, if not satisfied, yield anomalous comparative statics results, and thus seem at least to be necessary conditions for useful applications of the correspondence principle, at least in the context of analysis of the Walrasian system. Thus, even though Hicks' method seems to lack theoretical justification,7 the Hicks stability conditions produced by that method have proved exceedingly useful.

How can a wrong method yield useful results ? Leaving aside coincidence, the answer may lie in the two-way character of the correspondence principle. Samuelson [89] had observed that

not only can the investigation of the dynamic stability of a system yield fruitful theorems in statical analysis, but also known properties of a (comparative) statical system can be utilized to derive information concerning the dynamic properties of a system.

When Hicks is specifying the signs of changes in excess demands when a given price is put above or below its equilibrium value, various subsets of other prices remaining constant, he is at the same time implying specific comparative statics results. Provided these results correspond to known properties of a statical system the conditions implied should be related to stability conditions if the reciprocal character of the correspondence principle is valid.

Another reason why the Hicks method may appear more reasonable than Samuelson's original criticism of it suggests is that our knowledge of the precise laws governing dynamical systems is scanty. The empirical "output" according to the methodology of the correspondence principle is a set of comparative statics results, while the empirical "input" is (a) the nature of the dynamic processes, and (b) the assumption of stability. Acceptance of (b) is the essence of the correspondence principle, but how are we to determine (a)?

Consider, for example, the following alternative expressions for dynamical systems:

To each of these systems there will correspond a different set of stability conditions. System (1) is a version of that used by Samuelson to prove that the Hicks conditions are neither necessary nor sufficient for stability. Yet, as he himself noted, more complete generalizations such as (2) and (3) can be developed with different consequences for comparative statics. There is, therefore, an element of arbitrariness in the specification of dynamic systems in the absence of empirical information and there may on these grounds be a pragmatic justification for Hicks' method of developing " stability conditions " that are "timeless." The Samuelson criterion is completely general and is an appropriate methodological approach, but for purposes of yielding practical results generality often implies emptiness.

The purpose of this chapter is to show that the Hicksian stability analysis is a useful contribution to the integration of statical and dynamical theory. First, we shall show that the perfect and imperfect stability conditions do correspond to the dynamic stability conditions of some dynamic processes, irrespective of the pattern of signs of the price-matrix. Second, we shall argue that, despite their usefulness in the form Hicks presented them, the perfect stability conditions are not completely general, since they do not yield the information obtained by extending his method to the commodity "adopted" as the standard commodity. Third, we shall show that generalized conditions can be obtained by interpreting his device of holding subsets of prices constant with respect to the standard commodity as an arbitrary method of forming various composite commodity groupings. Further, we shall show that dynamic systems that fail to satisfy the generalized conditions will be unstable at some speeds of adjustment when a different commodity is adopted as the standard commodity in the dynamic system. And finally, we shall discuss the usefulness of the generalized Hicks conditions in devising dynamical rules for the hyperstability of " policy systems."8 The illustrative examples are all taken from the theory of foreign exchange markets, but the results, of course, apply to any generalized system.

The Hicks Conditions and Sliding Parities

Our first task is to show that the Hicks conditions do, in a sense, correspond to the conditions of convergence of some dynamic systems. Let us take as an example a problem in devaluation theory. We can describe a closed static equilibrium system of n + 1 currencies with prices (exchange rates) expressed in terms of currency 0, denoted by p1, . . . pn, as follows:

where Bi is the balance of payments of the ith country. In equilibrium each Bi = 0, while near the equilibrium we can write the system (4) as follows:

after expanding Bi in a Taylor series and omitting nonlinear terms.

Now let us suppose that the exchange rate of one country, say the rth country, appreciates in proportion to its balance of payments surplus according to the law

while all other exchange rates adjust instantaneously to equilibrium. The solution of the differential system (6) is

and Deltarr is the cofactor (principal minor) of the element in its rth row and rth column.

For the dynamic process implied in (7) to be stable it is necessary and sufficient that Delta / Deltarr < 0. But this condition is precisely (for the analogous problem in the Walrasian system) the Hicksian condition of imperfect stability for the rth currency; and when the method is applied to each currency (in succession, not simultaneously), we have the complete Hicksian conditions of imperfect stability:

A similar analysis can help to show the usefulness of the Hicks conditions of perfect stability. Suppose that one exchange rate, say pi, is held constant (relative to the numéraire). This amounts to dropping the ith row and column from Delta, so that if the original experiment were repeated, this time with the ith exchange rate constant, we would get, instead of (7),

and the stability condition Deltaii / Deltaii,rr< 0, which is one of the Hicksian conditions of perfect stability. Proceeding along these lines, holding one or another set of prices constant, we get the complete Hicks conditions of perfect stability.

But does this dynamic process have any economic plausibility ? Are we not, as Samuelson argued, allowing "arbitrary modification of the dynamical equations of motion"? The answer is, in a sense, yes. But this can be the exact method needed in the theory of policy, where our purpose is to design stable dynamic systems.

As an example, we might be interested in examining aspects of the stability of an exchange-rate system such as that recently advocated by sixteen distinguished academic economists [14] -a sliding parity system (with widened exchange-rate margins).9 Is it not precisely a set of conditions such as the Hicks conditions that would be involved? We might ask, first, what would happen if, say, Britain (which we shall identify with country 1) adopted a sliding parity system while a subset of other countries (2, . . ., j) allowed their exchange rates to float, and the remaining countries, k, . . ., n, kept their rates pegged to, say, the U.S. dollar (the currency of country 0). Then, if we suppose that balances of payments of countries whose rates float adjust instantaneously while the pound adjusts slowly, the path of the pound over time would be

for which knowledge of the Hicks conditions would be directly relevant. Thus the particular form of the dynamic system adopted -which countries are left out and which are left in- would depend on which of the Hicks conditions are satisfied. The Hicks method does, therefore, have a role to play in dynamic aspects of the theory of economic policy.

The Asymmetrical Position of the Standard Currency

The Hicks conditions, however, are not exactly what we need for the theory of economic policy, because, as we shall see, they are incomplete even in terms of Hicks' own method. In the experiments Hicks conducts to derive his stability conditions he accords the numéraire -the standard commodity- a special role. In this section we shall consider the precise deficiencies in the statical information provided by the Hicks conditions. This is best established by considering the comparative statics theorems implied by the Hicks conditions. Consider the equilibrium system

where, again, the Bi's are balances of payments, the p's are exchange rates, and alpha is a parameter.

Differentiation of (11) with respect to alpha yields

and the solutions for the exchange-rate changes are

Now consider an increase in demand for the currency of country i such that deltaBi / delta alpha > 0; while the excess demand for every other currency, at given exchange rates, is unchanged (deltaBj / delta alpha = 0 for i not=  j). Then, instead of (13), we have simply

By the Hicks conditions of imperfect stability Deltaii /Delta < 0, so deltaBi / delta alpha > 0 implies dpi / d alpha  > 0. Thus an increase in demand for the currency of the ith country raises the price of that currency after adjustment of all other exchange rates has been allowed for.

Similar implications follow from the conditions of perfect stability if we hold various subsets of other exchange rates constant relative to the numéraire. If, for example, the exchange rates of countries k, . . ., n are held constant, we get, instead of (14), the equation

the inequality being an implication of one of the conditions of perfect stability.

How can an increase in demand occur in a closed system ? Clearly only at the expense of other commodities (currencies) in the system. Cournot's law (or Walras' law in the context of the Walrasian system) ensures that

where the summation, it should be emphasized, extends over all the commodities. The interpretation of (14) is therefore that an increase in demand for (say) pounds (the currency of country i) at the expense of dollars raises the dollar price of the pound. Now if other exchange rates are held constant relative to the dollar, the proposition holds, if the Hicksian perfect stability conditions are satisfied, when the shift of demand is interpreted as being from the dollar and all currencies whose exchange rates are kept fixed to the dollar. Note, however, that the Hicks conditions do not give us the sign of

so that we cannot specify, on the grounds of the Hicks conditions alone, whether a shift of demand from dollars to pounds raises or lowers the price of (say) the franc relative to the dollar.

But now we are in a position to see the narrow form of the mathematical implications of the Hicks conditions. Consider a shift of demand from the franc (currency j) to the pound (currency i). Then deltaBs / delta alpha = 0 for s not= i, j, while deltaBi / delta alpha = - deltaBj / delta alpha >0 in view of (16); with no loss of generality we can make deltaBi / delta alpha = - deltaBj / delta alpha = 1. Substitution in (13) then gives the change in the dollar price of the pound and the franc:

The Hicks conditions do not provide us with the sign of either (18) or (19), nor, by analogy to (17), should we expect them to. But, by analogy with (14), we should expect the difference

to be unambiguous in sign for any system in which units are chosen so that each ps = 1, initially. When demand shifts from the franc to the pound, we should not expect to be able to predict the sign of the change in the dollar price of the pound or franc, but we should be able to determine, on the basis of the Hicks conditions, the sign of the change in the franc price of the pound, the expression given in (20). But the Hicks conditions are no help here, and this means that Hicks has not developed the mathematical implications of extending his method to the standard commodity.

The same information problem applies, a fortiori, when various subsets of prices are held constant. An implication of the Hicks condition of perfect stability is that a shift of demand onto pounds raises the price of the pound even when various currencies remain pegged to the dollar; this amounts to treating the dollar and the other currencies pegged to it as a composite currency. By analogy the price of the pound should rise when demand shifts from a currency other than the dollar, say, the franc, while other currencies (for example, the mark) are pegged to the franc.

Thus consider a shift of demand, at constant exchange rates, among three currencies i, j and k, such that

and every other deltaBr / delta alpha = 0. Then, from (13), we have

Applying the restrictions that and setting

we can deduce the change in price of the pound relative to the mark and franc.

where A is the sum of the cofactors of the elements in the following matrix:

The cofactor of, say, the element Deltajk can be related to the second cofactors of Delta by Jacobi's ratio theorem,

so that A /Delta can be written entirely as the sum of second cofactors, and (24) can be rewritten

The inequality sign should hold if an increase in demand for one country's currency occurs at the expense of any other country, one other currency price remaining constant relative to that country. But the mathematical information is not given to us by the Hicks conditions. The reason is that the mathematical implications of the Hicks method have not been developed with respect to the currency adopted as numéraire.l0

Generalization of the Hicks Conditions

When we do extend Hicks' method to make it " symmetrical " with respect to the numéraire (appreciating, say, the pound relative to, say, the franc, allowing various subsets of other currency markets to adjust), we get, of course, a set of conditions that specifies the signs of terms like those in (20) and (25). Along with the Hicks conditions, which can be written

[The next term requires that the ratio of the denominator of the second ratio and the sum of sixteen third minors be negative, and so on for successive ratios. The last term in the conditions of (27) specifies that the sum of the (n-l)th minors (n2 in number) be negative. But the (n-l)th minors are equivalent to the elements in the original determinant, so the last condition simply requires that the sum of all the elements in the original determinant Delta be negative.]

This suggests an alternative-and simpler-way of developing the generalized conditions. Consider the augmented determinant

formed by bordering Delta with its column and row sums, with a change of sign, so that

Then the extended Hicks conditions can be stated simply as the requirement that principal minors of B arranged in successive order oscillate in sign, except for the (singular) determinant B itself.11

Because the elements in the augmented determinant B are interdependent the generalized conditions can be expressed entirely in terms of the elements of the "normalized" determinant Delta. The supplemental conditions are

with the last condition reducing to the basic determinant Delta itself. These forms are equivalent to (30) and imply the signs of the ratios in (27).

This representation has the intuitive appeal of starting with the matrix of all the currencies in the system.l2 Thus, instead of omitting the numéraire currency at the outset, we start with a nonnormalized system of n + 1 currencies, exchange rates being expressed in terms of an abstract unit of account (for example, IMF par values), and apply the Hicks conditions allowing each currency the role of numéraire in turn.

Currency Areas

The above conditions are more general than the Hicks conditions. Yet they still do not exhaust the information inherent in the Hicks methodology. The Hicks method of holding various subsets of prices constant with respect to one another can be regarded as a device for constructing "composite commodities"; in the present context of currencies, we shall describe them as "currency areas." Now if we apply the Hicks method to a system based on arbitrary arrangements of countries in the currency areas, we get a further generalization of the results obtained by Hicks. When, for example, the mark is pegged to the dollar (the numéraire), the dollar and mark constitute a currency area. But there is no reason to restrict the formation of currency areas in a unidirectional attachment to the dollar. A group of currencies could be "attached" to the pound or the franc or any other currency, in principle. More importantly, we can then allow entire currency areas to appreciate and require that the balance of payments of the areas worsen, while various subsets of other currencies in the currency areas remain unchanged.13

The remarkable fact is that the conditions resulting from making arbitrary currency alignments among the (nondollar) countries and applying the Hicks method to the resulting matrix incorporate the conditions just developed as a special case. Thus consider the denominator of the first term in (27),

This term is the result of combining the ith and jth currencies together to form a currency area of those two countries. With no loss of generality we can write i = 1 and j = 2. Then if the first and second rows and columns are replaced by their combined rows and columns, we have

as can be proved by straightforward expansion. Similarly, it can be shown that the denominator of the second term in (27) is the determinant formed by replacing the ith, jth, and kth rows and columns of Delta by the amalgamated row and column.

When we now carry out Hicks' method for arbitrary arrangements of currency areas, extended over the whole range of currencies, we get a new set of conditions on the original (n x n) price matrix. These conditions can be expressed in a triangular arrangement of principal minors as follows:

The conditions on the left side of the stability triangle are the conditions of perfect stability Hicks developed; they do not provide the information implicit in extending the analysis to the numéraire commodity. The conditions on the base of the triangle result from extending the Hicksian method to the numéraire; they ignore the experiments resulting from allowing currency areas to appreciate. Finally, the conditions on the right side of the triangle are the conditions applicable when various sets of prices are raised in the same proportion, other prices remaining constant. More generally, the Hicks conditions on the left correspond to Hicksian adjustments when each currency (commodity) is treated in isolation; the adjacent conditions to their right are the Hicksian conditions when in the ith and ith goods move in the same proportion; and so on. The entire set of conditions are needed if the logic of Hicks' method is carried out to the bitter end.14

The General Conditions and Dynamic Stability

An important implication of the general conditions is that a system satisfying the Hicks conditions, but not the general conditions, will be stable or unstable depending on which currency is adopted as the key currency.15 Consider, for example, a world of three currencies, dollars (currency 0), pounds (currency 1), and francs (currency 2), and suppose that the balances of payments of the three countries are related to exchange rates according to the equations

where the exchange rates are defined in, and the Bi are expressed in, an abstract unit of account (IMF par value units). (Equilibrium exchange rates are unity or any multiple of unity, as the system is homogenous of degree 0.)

Let us consider a dynamic system in which the dollar is constant with respect to its par value so that the dollar becomes the effective numéraire. Let the par values of the pound and franc adjust in proportion to B1 and B2, respectively. We then have the following dynamic system:

In this system, the Hicks conditions of perfect stability, narrowly interpreted, are satisfied (since b11 = -2 < 0, b22 = -1 < 0), and

and the system is dynamically stable regardless of the (positive and finite) values of kl and k2 .

Consider, however, a system in which the par value of the franc is fixed so that it becomes the "key currency " instead of the dollar. The dynamic system then becomes

for which the Hicks conditions are not satisfied, It is dynamically stable or unstable according to whether k0 >< 4kl .

This result could be predicted at once by applying the general conditions as given in the stability triangle (32). The sum of the coefficients in (34a) are positive, so the general conditions are not satisfied.

The general conditions are necessary conditions for a system to be stable regardless of the currency (commodity) chosen as key currency (standard commodity) and regardless of how quickly the various exchange rates adapt to disequilibrium. This proposition is perfectly general in the sense that it is valid in the n-currency case.16

Hicksian Stability and the Theory of Economic Policy

I shall conclude this book by showing how the Hicks conditions, extended as above, can be useful in devising dynamic mechanisms that are "strongly stable." The problem could be approached from the direction of the correspondence principle, which, in a narrow version of it, suggests that we apply to comparative statics the conditions that the characteristic equation of the systems have negative real parts. The justification for this narrow version lies in the observation that the systems we know are not characterized by instability. But there is no reason, in principle, why stronger conditions could not be applied. We could, following Hicks (and Samuelson), require that the system be stable no matter which subsets of market variables are held constant. Alternatively, we could use conditions that the roots be real or complex according to whether we observe cycles in the system under investigation.17 If, for example, we observe an absence of cycles in the real world, we know at once that the Hicks conditions are sufficient conditions for dynamic stability.

In the theory of policy (under incomplete information) the problem is often to choose among different dynamic systems, or different degrees of centralization of a given dynamic system; this is often expressed in terms of allocating, dynamically, instruments to targets (the problem of effective market classification), and we may want to construct strongly stable systems: first, because systems near the borderline of stability may become unstable if disturbed by outside shocks; second, because the cost of adjustment may be higher if the system, even though stable, oscillates in its approach to equilibrium; third, because slight errors in manipulating rates of changes in instrumental variables (interest rates, exchange rates, and so on) may turn a weakly stable system into an unstable system; and, finally, because the time involved in approaching equilibrium may be less under strongly stable systems and rapid adjustment may be preferred to slow adjustment.

We can consider, therefore, the problem of choosing a dynamic control mechanism with "hyperstable" properties, in the sense that variables rise or fall whenever they are out of equilibrium, and show how the Hicks conditions can be of some help in constructing such a system when the precise location of an equilibrium is not known.

We take again as our example an international currency system. In a general hyperstable system it will be necessary to vary exchange rates, taking into account the balances of payments of each country. The problem is to find the weights each central bank should give their own balance of payments disequilibrium and that of the other countries.

Let Bi represent, as before, the balance of payments of the ith country, dependent upon the n exchange rates, according to the equation

that will make the system hyperstable. (The "speed" kij can be interpreted as the weight that country i has to give to the condition of the balance of payments of the jth country in adjusting its own exchange rate.)

It is readily shown that the system (36) is hyperstable if the k's are chosen so that

where alphai is a negative real constant and the Deltaji's are, as before, the cofactors of Delta. The condition (37) means that the hyperstable speeds are weighted elements of the inverse of the price matrix.

To prove this proposition we need to prove that the dynamic system

But, from the properties of any determinant, the typical term

has a value of unity for k = i and a value of zero for k not= i. The system (39) therefore reduces to

It is instructive to write out the hyperstable system (38) in detail to see clearly the implications of Hicksian perfect stability for dynamics:

We shall also find it convenient to consider a reduced system in which we choose the alphai, the rate at which each pi is restored to equilibrium (with a negative sign), to be equal to the corresponding Hicksian conditions of imperfect stability as given in (9), that is, we equate

Two observations can immediately be made about (44). First, if the elements of the inverse matrix all have the same sign, hyperstability implies that positive weights be assigned to each balance of payments. Thus, "Britain" should depreciate (appreciate) more rapidly the greater the deficits (surpluses) in the balance of payments of other countries, for any given deficit in her own balance (this implies corresponding changes in the U.S. balance). But from Mosak's theorem the elements in the inverse will all have the same sign if the original currency matrix [bij] is a gross substitute matrix provided [bij] is Hicksian; and it will be Hicksian provided the dollar is also (reciprocally) a substitute for all other currencies.

The second point to notice about (44) is that " Britain" should attach less weight to the balance of payments of other countries' currencies than to her own balance if currencies are all substitutes for one another; this follows because every ratio Deltaji / Deltaii < 1 for j not= 1.19

Leaving now the special case of the gross substitutes to return to the more general case represented by equations (42), we can find immediate implications of the Hicks conditions. First, if the Hicks conditions are satisfied, " normal" adjustments are implied in the sense that, ceteris paribus, a deficit in a country's balance of payments suggests depreciation and a surplus appreciation; this follows because every kii = Deltaii /Delta > 0 given the Hicks conditions of imperfect stability and alpha i< 0.

But more than this can be said. The following identities have to hold, from our definitions and Jacobi's ratio theorem:

the inequality following at once from the Hicks conditions of perfect stability. Proceeding to the last term we have

More generally, if the basic matrix [bij] is Hicksian, the matrix of the speeds required for hyperstability must satisfy the conditions that every principal minor be positive, that is,

Analogous conditions hold for the extended Hicks conditions if the system is to be hyperstable regardless of the currency used as the key currency. In this sense the Hicks conditions alone are sufficient to establish in a weak sense the correctness of exchange rate policies directed at correcting "own" balances of payments.

These developments conform to the conclusions of economic intuition indeed, they may be interpreted as bringing the mathematical treatment of the subject closer to the level of common sense. They nevertheless suggest that the Hicksian stability analysis is not lacking in significance for the integration of dynamical and statical theory.20, 2l

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Notes

1 Adapted from: a forthcoming article in Essays in Honor of Sir John R. Hicks (N. Wolfe, ed.).

2 I am happy to acknowledge many helpful conversations on the subject matter of this chapter with J. C. Weldon of McGill University.

3 Strictly, the problem is not trivial even in the case of exchange of two commodities; first, because of complications associated with the possibility of time derivatives of various orders of price changes entering the excess demand (X) functions, in a system such as second, because of nonlinearities in the excess demand function; and, third, because market exchange of two commodities among many people may involve adjustments of quantities toward budget constraints, giving rise to the more complicated dynamics such as Marshall postulated in his foreign trade analysis.

4 Let Xi = Xi(p, . . ., pn) be the excess demand functions for commodities i = 1, . . ., n. By differentiation . Solving for the dpi we get , where Delta is the determinant of the system and Deltaji, its first cofactors. When all other prices adapt so that every dXj = 0 except dXi, we have the solutions dpi = (Deltaii /Delta) dXi; if various prices k, . . ., n, are held constant with respect to the numéraire we get instead

Imperfect stability requires that Deltaii /Delta < 0 for i = 1, . . ., n, whereas perfect stability requires that Deltaii,kk, ... ,nn / Delta kk ... ,nn < 0 for any subset of commodities k, ...,n, excluding commodity i. The Hicksian perfect stability conditions are thus

the first condition being the condition of imperfect stability.

5 It is slightly ironic that Marshall in his 1879 manuscript on foreign trade utilized the link between dynamics processes and stability, but not explicitly the link between stability and comparative statics, whereas Hicks utilized the link between stability and comparative statics but not the link between dynamics and stability. Samuelson used both links in his integration of dynamics and statics.

6 Actually, Metzler's example (p. 285) that the Hicks conditions are not sufficient for stability to be independent of the speeds of adjustment contains a numerical error that mars his demonstration [the second cubic of his footnote 12 should read lambda3 + 4lambda2 + 3.4lambda + 13.2 = 0 instead of lambda3 + 4lambda2 + 2.6lambda + 13.2 = 0 and in the first (correct) cubic the Routh conditions are satisfied]. But a slight adjustment to his counterexample can nevertheless demonstrate his point. If, for example, speeds are chosen so that, in his terminology, kl = k3 = 1, giving the cubic lambda3 + (2 + k2)lambda2 + (1+ 1.2k2) lambda + 6.6k2 = 0, the Routh conditions will not be satisfied for values of k2 somewhat less than 1.

7 In this connection Samuelson writes ([88], p. 554): "In principle the Hicks procedure is clearly wrong, although in some empirical cases it may be useful to make the hypothesis that the equilibrium is stable even without the 'equilibrating action' of some variable which may be arbitrarily held constant." Samuelson provides an example in connection with the Keynesian system later in his paper.

8 See, for example, Mundell [73] and the references there to the principle of effective market classification.

9 See page 111 for a list of economists who signed the petition.

10 This is perhaps apparent immediately, since the Hicks conditions can be expressed in terms of the commodity chosen as numéraire, the excess demand for which is never allowed to go to zero. On this point (but without apparent recognition of its implications) see Lange ([41], p. 92).

11 More compactly, we can state the conditions as requiring that every first minor of B be "Hicksian." Not all the conditions are independent, however, since B00 = B11 = Bnn in view of the characteristics of B.

12 One might think that Hicks really intended his conditions to extend over the entire range of excess demand coefficients (including the numéraire), this would be incorrect since the last (augmented) determinant is singular. Alternatively, it might be thought that Hicks intended the conditions to apply no matter what commodity were taken as numéraire. The latter interpretation seems to be fortified by a footnote which states ([24] p. 75): " [this] can be seen at once if we adopt the device of treating X (momentarily) as the standard commodity, and therefore regarding the increased demand for X as an increased supply of the old standard commodity m ...." If this device were adopted to derive the Hicks conditions it would indeed result in conditions equivalent to the above stability conditions. Yet this interpretation would conflict, not only with all subsequent interpretations of the Hicks conditions by other writers on stability, but also with Hicks' own explicit statement in the Mathematical Appendix (p. 315), which specifies that the conditions must hold "for the market in every Xr (r = 1, 2, 3, . . ., n-1)"; that is, the market for the numéraire (the nth commodity) is omitted, In any case his discussion on pages 68-71 fails to make the point clear, while his third graph in Figure 16, which he asserts is stable, is actually totally stable only if the line along which there is zero excess demand for the standard commodity (not drawn in his figures) is inelastic referred to the abscissa. The discussion is not completely clear, however, and I am therefore inclined to interpret Hicks as actually intending to give full coverage to the numéraire but not completing the mathematical implications of doing so.

13 I have considered the problem of " optimum" currency areas in terms of its attributes for stabilization policy, optimum exploitation of the functions of money, and so on in Chapter 12; the present analysis suggests an additional criterion in terms of stability conditions.  

14 Stability conditions should, in principle, be "invariants" in the sense that they are independent of the choice of units in which commodities are measured; this corresponds to Lange's "principle of invariance" ([41], p. 103). The above stability conditions are invariants only if, as is assumed in the case of the foreign exchange market analysis, the bij's are all measured in equivalent currency units.

In the general case, however, where prices respond to excess demands denominated in physical quantity units, the units in which bij( j = 1, . . ., n) is measured differ from the units in which bkj(j = 1, . . ., n) is measured. This means that invariant stability conditions require that each row of every stability determinant involving the sum of elements be multiplied by an arbitrary number reflecting units of measurement. Thus "unit-invariant" stability in the general case involves the sign of terms such as

For arbitrary values of ki and kj extended over the entire range of the (n+ 1) x (n+ 1)-B determinant, and given the homogeneity postulate, the general conditions then imply that all goods are gross substitutes.

15 From a historical point of view, it is somewhat amusing that although the Hicks conditions are not symmetrical with respect to the commodity chosen as numéraire, neither are the (normalized) dynamic systems used to refute the validity of the Hicks conditions as true dynamic stability conditions.

16 The proof in the general case follows by analogy to the proof of Metzler ([62], pp. 280-285) when his method is extended to include the augmented system.

17 I have applied this method to the problem of disentangling lags in expectational and cash balance adjustments [77].

18 More directly the matrix equationif k = B-1.

The usefulness of this result lies not so much in providing a policy maker with the rules of adjustment, since there is no problem if the basic matrix is known, and the rule gives insufficient information if the basic matrix is not known. The point is rather that pieces of information about the inverse may be sufficient to distract policy makers from mistakes, and that conditions like the Hicks conditions may be a sufficient guide to the relative importance to be attached to particular instruments.

19 I have discussed the implications of this condition in some detail [78].

20 Even the gap between the true dynamic stability and Hicksian stability may be bridged by integrating Hicksian stability conditions with the "Samuelson-Le Châtelier principle" (see, for example, Samuelson [93]) which can be expressed entirely in terms of the Hicksian determinants. Hicksian stability and dynamic stability mesh together under appropriate conditions of gross substitutes, gross complements, and symmetry, a common link being sign symmetry; and sign symmetry of the inverse of the basic Hicksian matrix is sufficient for at least one of the Le Châtelier conditions to hold (DeltaDeltaii,jj - DeltaiiDeltajj < 0 if Deltaij and Deltaji, have the same sign).

21 A further implication of the Hicks conditions was called to my attention by Daniel McFadden, who had proved, in a paper presented at the 1963 Econometrica Society meetings in Boston, Massachusetts, that if the Hicks conditions of perfect stability are satisfied, a stable dynamic system of the form can always be found for a diagonal K matrix with positive diagonal coefficients; the result also holds in the global form. Prior to McFadden's result, and unknown to him, a local version of the theorem had been published; see Fisher and Fuller [15].

In the context of the currency problem discussed above, the theorem means that if the Hicks conditions are satisfied, it is always possible to find a stable dynamic system in which exchange rates are adjusted to " own " balances of payments only.

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Literature Cited

[14] W. J. FELLNER et al., Maintaining and Restoring Balance in International Payments. Princeton, N.J.: Princeton University Press, 1966.

[15] M. E. FISHER and A. T. FULLER, " On the Stabilization of Matrices and the Convergence of Linear Iterative Processes," Proc. Cambridge Phil. Soc., 54 (1958).

[24] J. R. HICKS, Value and Capital. 2nd ed. Fair Lawn, N.J.: Oxford University Press, 1946.

[41] O. LANGE, Price Flexibility and Full Employment. Bloomington: Indiana University Press, 1944, p. 92.

[62] L. A. METZLER, "Stability of Multiple Markets: The Hicks Conditions," Econometrica, 13, 277-292 (Oct. 1945).

[73] R. A. MUNDELL, "The Appropriate Use of Monetary and Fiscal Policy for Internal and External Stability," IMF Staff Papers, 9, 70-79 (March 1962).

[77] R. A. MUNDELL, "Growth, Stability and Inflationary Finance," Jour. Pol. Econ.. 73. 97-100 (April 1965).

[78] R. A. MUNDELL, "The Significance of the Homogeneity Postulate for the Laws of Comparative Statics," Econometrica, 33, 349-356 (April 1965).

[88] P. A. SAMUELSON, "The Stability of Equilibrium: Comparative Statics and Dynamics," Econometrica, 9, 97-120 (April 1941).

[89] P. A. SAMUELSON, " The Stability of Equilibrium: Linear and Non Linear Systems," Econometrica, 10, 1 (April 1942).

[93] P. A. SAMUELSON, " An Extension of the Le Châtelier Principle," Econometrica, 28, 368-379 (April 1960).


© Copyright Robert A. Mundell, 1968